Irrationals Do … What? (Warning: Graphic!)

As I see it, this is a very graphic demonstration of how irrational numbers relate to rational numbers. It definitely concerns some inherent property of rational numbers. The problem is, I really don’t know what that property is or how to describe it.

We start with simple periodic functions like sine and cosine. Being periodic, they repeat with a certain period : ) But that period can be controlled. Below, Cos(x), Cos(3/2*x), and Cos(2*x) are all plotted on top of each othe, the first in blue, second in red, third in yellow-ish. As x increases, the functions oscillates and eventually repeat.

The Clearly, Cos(2*x) oscillates twice as fast as Cos(x) and has a period half as long.

Basic stuff, but then we start combining things.

Consider the set of points of the form {Cos(t), Sin(t)}. As t goes from 0 and increases, the x-value changes with cosine and the y-value with sine. And the points trace out the following picture.

A circle. But the important thing is that it is a closed figure. If we keep increasing t, we’ll be tracing out the same circle over and over again. This is because the x-values have a period in t, and the y-values have a period in t, and any time these periods align, we begin to repeat what we’ve already drawn.

Changing the coefficients on t produce a range of images, though after a while they all look the same.

{Cos(3 t), Sin(t)} produces the following.

{Cos(3 t), Sin(5 t)} produces this.

And for a slight change, {Cos(19 t), Cos(20 t)} produces this beast.

It just gets crazier. But the point is that in any case, because the x-values and the y-values have discrete periods, eventually, when you start running period after period after period, those periods line up and you begin to repeat the process. Like if the period of one were based on 3, and the period of the other based on 5, the figure would begin to repeat every 15. In all these cases, the periods come into alignment over time, and the process, and thus figure, begins to repeat.

But this is the property of rationals I was referring to, the one I couldn’t name. Given two rational numbers, you can add the first enough times, and the second enough times, to produce equal quantities. Like, for rational numbers a and b, you can find integers j and k such that a*j = b*k.

But, the relationship to the picture is that the periods of all the functions we’ve examined so far is that they relate to each other -rationally-. The period of one will be 20/19′ths the period of the other, etc.

And everything is blown to hell when you introduce irrationals. If we relate the periods of the two, say make one times the period of the other…

Observe, {Sin(t), Cos( t )}, and this is running t from 0 to 30.

Notice that it lacks the very clear symmetry that characterized the first few graphs. However, maybe we just didn’t trace out the figure long enough? This is t running from 0 to 60.

Still lacks the same kind of symmetry, and notice the way the curve just sort of ends down near the bottom left. It still hasn’t begun to loop or repeat as the others have. So we run t even further, from 0 to 1000.

At this point, it should be clear that the curve isn’t repeating. In fact, it looks as though if it runs long enough, the curve will hit every single point in the square, covering it completely.

This is all due to the fact that the periods of the x-values and y-values no longer relate rationally. Because one is based on an irrational number, running the figure for longer and longer times, the two periods never line up to begin repeating. The figure changes continually, filling in the square as above. Using the same description as above, it would be like finding integers j and k such that j = k*. But of course, that would yield , and as the square root of 2 is irrational, that never will happen.

This is certainly a very dramatic representation of something – though I just lack the words to name it.

Interestingly though, think about it like this – suppose you had a system of some variety, mathematical, mechanical, programmatical, what have you, that was dependent on the values of a few parameters. What this demonstrates is that you can get radically different behaviors by changing the -type- of value you give it. 1.4 might yield some kind of interesting result, and 1.415 might yield a slightly different, albeit related behavior, but as the value swept between the two, and hit , it might just seem to explode with crazies.

So, the effect of number type vs number value. Interesting.